Are asylum seekers illegal immigrants

No, if you are asylum seeker and your case in the process - YOU ARE NOT ILLEGAL. When I applied to USCIS for political asylee status, it took 7 weeks to get granted approval status. I was lucky and get that without problems and time delays.

But, if they transfer your case to Immigration Court, it takes additional time from 6 months to several years. And you may get permit to work and Social Security, it depends on them. So, you can work and wait for your decision. But, if you don't get permit to work and SSN, you still legally in the country and can not be deported until final immigration decision.